Saturday, February 11, 2012
Numbers
Nice and bright along Horton Lane this morning where a dozen or so large thrush like birds were flitting through the hedgerow trees alongside the golf club. Big things, bigger than a blackbird but smaller than a wood pigeon, grey brown on top and pale underneath, not hugely speckled as far as I could see. Back at the RSPB identification widget to decide that the they were probably missel thrushes. Only weak part of the identification being the absence of conspicuous speckles.
Further along the lane, I got onto wondering again about large numbers (see January 23rd). First thought was that one needed some rules. Rule 1, the number has to be a large positive integer. Rule 2, the number has to be a property of some object or situation. Rule 3, determination of the number has to be replicable. One might devise further sub-rules about what constituted admissible replication in this context. Different times? Different people? Different places?
An example of what I mean by rule 2 would be to say that the target number is the number of prime numbers less than 10^100. With a bit of effort one could come up with an upper bound for such a target number that one could compute in reasonable time. Which might prompt one to add a rule 4, that determination has to take less than some designated time. Or a rule 5, that the equipment required to make the determination should not cost more than so many dollars. Or pounds sterling if you prefer.
Another scientific number might the number of carbon atoms in a molecule of some known protein. Which would be OK but probably not very large. Or the number of base pairs in some particular chromosome. Which I suspect might be larger but not so OK. One cannot rely on a chromosome always having the same number of base pairs. Bits might get missed off or left off the ends.
More mundane, one might have a carefully made, sand tight box, perhaps made of stainless steel with welded and polished joins. One then fills it up with a coarse washed sand - the idea of the washing being to remove all the smaller grains - and counts the grains. If one allowed a grain of sand to the cubic millimetre, a cubic metre of sand would contain around 1,000 million of them. A respectable 10^9. But I suspect there would be difficulties at the margin. Some of the grains would fall apart during the counting process, or perhaps disintegrate more or less completely. And then,what sort of machine would be needed to count the grains in such a way that they could be counted again? How would you stop the odd grain lodging in the mechanism?
Next thought was Lego. You could have a sealed box containing some known number of Lego bricks, perhaps direct from the manufacturer. You then make a mound of such boxes. All you have to do is count the boxes. Could you be sure that among a million boxes - assuming you could lay your hands on that many - that there were not going to be some duds? A spot check of boxes would not really amount to determination. And how do you spot check boxes in large numbers while remaining sure that none of the bricks are leaking out? Remember all the fuss about hanging chads in the Florida part of a recent presidential election.
I think that one of the triggers for all this was reading in Yau about a breed of geometer who make a living by devising complex surfaces and then counting the number of different, infinite straight lines which can be placed on that surface. It seems that if you get the surface right, the number is both finite and large. And determinate. So perhaps the criterion for success should be informal rather than rule driven. The panel of judges just have to like the thing, to find the number and its determination attractive.
Perhaps set up a challenge cup with a prize of £100,000 every fourth year. The sort of thing that some slightly eccentric billionaire might put up the dosh for?
Further along the lane, I got onto wondering again about large numbers (see January 23rd). First thought was that one needed some rules. Rule 1, the number has to be a large positive integer. Rule 2, the number has to be a property of some object or situation. Rule 3, determination of the number has to be replicable. One might devise further sub-rules about what constituted admissible replication in this context. Different times? Different people? Different places?
An example of what I mean by rule 2 would be to say that the target number is the number of prime numbers less than 10^100. With a bit of effort one could come up with an upper bound for such a target number that one could compute in reasonable time. Which might prompt one to add a rule 4, that determination has to take less than some designated time. Or a rule 5, that the equipment required to make the determination should not cost more than so many dollars. Or pounds sterling if you prefer.
Another scientific number might the number of carbon atoms in a molecule of some known protein. Which would be OK but probably not very large. Or the number of base pairs in some particular chromosome. Which I suspect might be larger but not so OK. One cannot rely on a chromosome always having the same number of base pairs. Bits might get missed off or left off the ends.
More mundane, one might have a carefully made, sand tight box, perhaps made of stainless steel with welded and polished joins. One then fills it up with a coarse washed sand - the idea of the washing being to remove all the smaller grains - and counts the grains. If one allowed a grain of sand to the cubic millimetre, a cubic metre of sand would contain around 1,000 million of them. A respectable 10^9. But I suspect there would be difficulties at the margin. Some of the grains would fall apart during the counting process, or perhaps disintegrate more or less completely. And then,what sort of machine would be needed to count the grains in such a way that they could be counted again? How would you stop the odd grain lodging in the mechanism?
Next thought was Lego. You could have a sealed box containing some known number of Lego bricks, perhaps direct from the manufacturer. You then make a mound of such boxes. All you have to do is count the boxes. Could you be sure that among a million boxes - assuming you could lay your hands on that many - that there were not going to be some duds? A spot check of boxes would not really amount to determination. And how do you spot check boxes in large numbers while remaining sure that none of the bricks are leaking out? Remember all the fuss about hanging chads in the Florida part of a recent presidential election.
I think that one of the triggers for all this was reading in Yau about a breed of geometer who make a living by devising complex surfaces and then counting the number of different, infinite straight lines which can be placed on that surface. It seems that if you get the surface right, the number is both finite and large. And determinate. So perhaps the criterion for success should be informal rather than rule driven. The panel of judges just have to like the thing, to find the number and its determination attractive.
Perhaps set up a challenge cup with a prize of £100,000 every fourth year. The sort of thing that some slightly eccentric billionaire might put up the dosh for?
# posted by Jim @ 7:39 AM 
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16/2/2012: I have now been told that the birds were more likely to have been fieldfares. Partly on the grounds that missel thrushes do not flock and partly on the grounds that the speckles on fieldfares are less conspicuous than those on missel thrushes. Looking at the pictures on Google this seems a reasonable point.
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